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In our earlier work, we have measured human intelligibility of stimuli reconstructed either from the short-time magnitude spectra or short-time phase spectra of a speech signal. We demonstrated that, even for small analysis window durations of 20-40 ms (of relevance to automatic speech recognition), the short-time phase spectrum can contribute to speech intelligibility as much as the short-time magnitude spectrum. Reconstruction was performed by overlap-addition of modified short-time segments, where each segment had either the magnitude or the phase of the corresponding original speech segment. In this paper, we employ an iterative framework for signal reconstruction. With this framework, we see that a signal can be reconstructed to within a scale factor when only phase is known, while this is not the case for magnitude. The magnitude must be accompanied by sign information (i.e., one bit of phase information) for unique reconstruction. In the absence of all magnitude information, we explore how much phase information is required for intelligible signal reconstruction. We observe that (i) intelligible signal reconstruction (albeit noisy) is possible from knowledge of only the phase sign information, and (ii) when both time and frequency derivatives of phase are known, adequate information is available for intelligible signal reconstruction. In the absence of either derivative, an unintelligible signal results.